Electrical – the minimum required distance from a shower to an electrical outlet

There is no minimum distance specified in the NEC. It can be as close as is physically possible given the constraints of framing, trim, etc. There's several rules about maximum distance and spacing and when to include door swings or not, and what constitutes a usable wall requiring outlets in the first place. But no minimum.

Now, conceivably, someone could interpret that since the outlet is within 6 feet of a "damp" area (patio) that it may need to be GFCI protected. I personally believe this stretching the intent of the NEC too far since the theoretical appliance in a damp area would have it's cord running through an open exterior door into a different room.

There could be other requirements by Washington state or your local jurisdiction. The NEC is mute on this matter, but you should inquire with your local building authority to be absolutely sure there are no other applicable requirements or "creative" interpretations of GFCI requirements.

The real answer is "it depends on what else that circuit does". According to the NEC, if your outlet circuit goes to more than one bathroom, powering the fan/light from it is a violation. If it's only in that bathroom, you may be able to do this without violating code, but I'd recommend consulting a licensed electrician for advice, if nothing else.